Finitistic dimension and Endomorphism algebras of Gorenstein projective modules

TL;DR

This paper investigates the relationships between the homological dimensions of an Artin algebra and its endomorphism algebra of a Gorenstein projective module, providing bounds based on the module's properties and flat dimension.

Contribution

It characterizes the homological dimensions of endomorphism algebras of Gorenstein projective modules over CM-finite and Gorenstein algebras, extending understanding of their finitistic and global dimensions.

Findings

Bound on fin.dim B in terms of fin.dim A and module properties

Bound on gl.dim B for Gorenstein algebras with n ≥ 2

Use of restricted flat dimension to relate homological dimensions

Abstract

Let $A$ be an Artin algebra, $M$ be a Gorenstein projective $A$-module and $B =$ End$_A M$, then $M$ is a $A$-$B$-bimodule. We use the restricted flat dimension of $M_B$ to give a characterization of the homological dimensions of $A$ and $B$, and obtain the following main results: (1) if $A$ is a CM-finite algebra with $\cal GP$($A$) = add$_AE$ and fin.dim $A \geq 2,$ then ${\rm fin.dim}\ B \leq {\rm fin.dim}\ A + {\rm rfd}(M_B) +{\rm pd}_B{\rm Hom}_A(M, E);$ (2) If $A$ is a CM-finite $n$-Gorenstein algebra with $\cal GP$($A$) = add$_AE$ and $n \geq 2$, then gl.dim $B \leq n + {\rm pd}_B{\rm Hom}_A(M, E).$